Group having 3 proper subgroups

Feb 2009
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10
Chennai
What can we say about a group which has exactly 3 proper subgroups?
 
Oct 2009
4,261
1,836
What can we say about a group which has exactly 3 proper subgroups?


Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

Tonio
 
May 2009
1,176
412
Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

Tonio
No - proper means it is properly contained in it. That is to say, \(\displaystyle H \lneq G\). So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).
 
May 2009
1,176
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As tonio said, a cyclic group of order \(\displaystyle p^3\) can be one of examples, where p is a prime number.

See my previous post here .
As a warning to the OP, this does not classify them all. Another example would be \(\displaystyle V_4\), the klein 4-group (or, more generally, the cross-product of two groups of prime order).
 
Oct 2009
4,261
1,836
No - proper means it is properly contained in it. That is to say, \(\displaystyle H \lneq G\). So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).

Yes. I oversaw the word "proper" in the OP, but still the hint remains...though nevertheless there are OTHER examples: for example, a cyclic group of order 10...

Tonio